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Zaare
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How can I show that if [tex]\phi(t)[/tex] is a characteristic function for some distribution, then [tex]|\phi(t)|^2[/tex] is also a characteristic function?
I think I can show this, but how does that help me?matt grime said:Let Y be another r.v. with distribtuion as -X, then its characteristic function out to be the complex conjugate of phi, I think. At least that seems plausible.
X and Y have to be independent for this to be ture, right? And I have to show that this is true for X and X even though X and X are not independent.matt grime said:Given two distributions X and Y, the char function of their sum is the product of their char functions.
matt grime said:Let Y be another r.v. with distribtuion as -X, then its characteristic function out to be the complex conjugate of phi, I think. At least that seems plausible.
A characteristic function is a mathematical function that describes the probability distribution of a random variable. It is defined as the expected value of the complex exponential function of the random variable. In simpler terms, it is a function that uniquely identifies a probability distribution.
A characteristic function and a probability density function (PDF) are closely related. The characteristic function is the Fourier transform of the PDF. This means that the characteristic function contains all the information about the PDF, such as the mean, variance, and higher moments.
One advantage of using characteristic functions is that they can be used to find the probability distribution of a sum of independent random variables. This is known as the characteristic function of a sum property, and it makes calculations involving multiple random variables much easier.
Yes, the shape of a characteristic function can be used to identify the type of probability distribution. For example, a Gaussian distribution will have a characteristic function with a bell-shaped curve, while a uniform distribution will have a characteristic function with a constant value.
One limitation of using characteristic functions is that they may not exist for all probability distributions. Some distributions, such as the Cauchy distribution, do not have a characteristic function. Additionally, calculating a characteristic function may be computationally intensive for complex probability distributions.